\[ \int \frac {x^3}{\sqrt {a x^2+b x^3+c x^4}} \, dx \]
Optimal antiderivative \[ \frac {\left (-4 a c +3 b^{2}\right ) x \arctanh \left (\frac {2 c x +b}{2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}}\right ) \sqrt {c \,x^{2}+b x +a}}{8 c^{\frac {5}{2}} \sqrt {c \,x^{4}+b \,x^{3}+a \,x^{2}}}+\frac {\sqrt {c \,x^{4}+b \,x^{3}+a \,x^{2}}}{2 c}-\frac {3 b \sqrt {c \,x^{4}+b \,x^{3}+a \,x^{2}}}{4 c^{2} x} \]
command
integrate(x^3/(c*x^4+b*x^3+a*x^2)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {1}{4} \, \sqrt {c x^{2} + b x + a} {\left (\frac {2 \, x}{c \mathrm {sgn}\left (x\right )} - \frac {3 \, b}{c^{2} \mathrm {sgn}\left (x\right )}\right )} + \frac {{\left (3 \, b^{2} \log \left ({\left | -b + 2 \, \sqrt {a} \sqrt {c} \right |}\right ) - 4 \, a c \log \left ({\left | -b + 2 \, \sqrt {a} \sqrt {c} \right |}\right ) + 6 \, \sqrt {a} b \sqrt {c}\right )} \mathrm {sgn}\left (x\right )}{8 \, c^{\frac {5}{2}}} - \frac {{\left (3 \, b^{2} - 4 \, a c\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{8 \, c^{\frac {5}{2}} \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {x^{3}}{\sqrt {c x^{4} + b x^{3} + a x^{2}}}\,{d x} \]________________________________________________________________________________________